The newly minted Confederacy was only worried about preserving slavery and the stiffly ranked society that slavery created -- but in Lincoln's mind the issue was even larger: Secession was anarchy -- and no friend to democracy.
On the day Lincoln was inaugurated in 1861, the United States was the only large-scale example of a democracy left in the world. There had been others -- but from 1789 until 1861, all of those other struggles for democracy had been snuffed out by kings, emperors and dictators of various sorts.
If the American democracy shattered itself because seven states weren't willing to abide by the outcome of the presidential election, then every one of those kings, emperors and dictators would be able to say to their nations, "See what democracy gets you? Instability. Disorder. The moment one faction loses out to another, they want to break the whole thing up! That's democracy!"